(version en francais)

     Throughout the questions, V is supposed to be a n-dimensional vector space on the Field K (which will always be R or C). A subset F of L(V), set of endomorphisms of V is said to be triangulable if there exist a basis in V in which the matrix of ANY element of F is upper triangular. It is reminded that a subspace W of V is said stable by F if, for all u Î F, W is stable by u, i.e. u(x) Î W for all x Î W.

     The main topic of this problem consists in looking for eigenvectors that are common to all elements of a subset F of L(V), endowed with suitable properties, the main application being to obtain some sufficient conditions for triangulation.

     We assume in the remaining of this part, that F is a subset of L(V) such that, for any u Î F and v Î F, one have u° v = v ° u. The purpose of the following questions is to show that F is triangulable.
2^°) Let u Î F, l is a eigenvalue of u and let V_u(l) be the corresponding eigenspace. Show that V_u(l) is stable by F.
3^°) Show that the elements of F have a common eigenvector.
4^°) Show that F is triangulable.
5^°) We additionaly assume that any element of F is diagonalizable. Can we find a basis in V in which the matrix of any element of F is diagonal?
6^°) Re-do the problem raised in question 2^°, replacing C by R.

     Given u Î L(V) and v Î L(V), let [u,v]=u° v - v° u. A subset F of L(V) is called a Lie algebra (made of endomorphisms of V) if the following conditions are fulfilled :

(i)  F is a vector subspace of L(V)
(ii)  For any u Î F and v Î F, [u,v] Î F.

     We will call dimension of a Lie algebra F, which will be denoted by dim(F), its dimension as a subspace over the field K.

     Let F be a Lie albegra, we call ideal of F any subspace I of F such that [u,v] Î I for any u Î F and v Î I.
1^°) Let F be a Lie algebra of dimension 2, such that there exists u₀ Î F and v₀ Î F obeying [u₀,v₀] ¹ 0 . Let also F¢ ba another Lie algebra of dimension 2, obeying the same property. Show that there exists an isomorphism (vector space isomorphism) f from F to F¢ such that f([u,v])=[f(u),f(v)] for any u Î F and v Î F.

     Let F be a Lie algebra and I an ideal of F. Given a linear form l on I, we denote by W the subspace of V of vectors x such that v(x)=l(v)x for any v Î I. The purpose of questions 2^° to 5^° is to show that W is stable by F.

     Let u Î F, and x be a non-vanishing element of W ; we define by recursion a sequence (x_k) in the following way: x₀=x and x_k = u(x_k-1) for any integer k ³ 1.
2^°) Show that, for any k Î N and any v Î I, v(x_k)-l(v)x_k lies in the subspace generated by {x₀,x₁,¼,x_k-1}.
3^°) Let U be the subspace of V generated by the vectors x_k, where k is any positive integer. Show that U is stable by I È u.
4^°) Give a relation between l([u,v]) and the trace (i.e. the sum of eigenvalues) of the restriction to U of the endomorphism [u,v].
5^°) Show that W is stable by F.

     A Lie algebra F is said to be solvable if their exists an increasing sequence

{0} = F0 Ì F1 Ì ¼ Ì Fp = F

of subspaces F such that, for any integer k obeying 1 £ k £ p, one has :

[u,v] Î Fk-1 for any u Î Fk and v Î Fk

6^°) Show that any Lie algebra of dimension £ 2 is solvable.

     The purpose of the following questions is to proove the "Lie Theorem" which states that any solvable Lie algebra is traingulable. We consequently let F be a solvable Lie algebra.
7^°) Let d=dim(F). Show that there exists an ideal I of F, of dimension d-1. Show that I is also a solvable Lie algebra.
8^°) Show that the elements of F have a common eigenvector.
9^°) Show that F is triangulable.
10^°) Show that, conversely, any triangulable Lie algebra is solvable.
11^°) Show that the result of question I.4^° is a corrolary of the "Lie Theorem".

     In this part, the field K can be either R or C. For any u Î L(V), we will denote by ad_u the element of L(L(V)) defined by ad_u(v)=[u,v] for any v Î L(V).
1^°) Show that, for u Î L(V) and v Î L(V), we have ad_[u,v]=[ad_u,ad_v].
2^°) Show that, if u is a nilpotent operator in L(V), then ad_u is a nilpotent operator in L(L(V)).
3^°) Let F and G be two Lie algebras (made of endomorphisms of V) such that G Ì F. Let H be a complementary subspace of G in F, and let q be the projector on H parallel to G, i.e. the application from F to H which maps any u Î F to the unique v Î H such that u-v Î G. Show that there exists one and only one linear application

p: G ® L(H)

such that, for any g Î G and any u Î F,

p(g)(q(u)) = q([g,u]).

     We now denote by F a Lie algebra (made of endomorphisms of V), such that any element of F is a nilpotent operator in V. The purpose of the following questions is to demonstrate the "Engel's theorem", which states that their exists a non vanishing vector x Î V such that u(x)=0 for any u Î F.
4^°) Let G be a second Lie algebra (made of endomorphisms of V), such that G Ì F and G ¹ F. We keep the same notations as in the previous question and we let F¢ = p(G), V¢=H. Show that F¢ is a Lie algebra ( made of endomorphisms of V¢), that dim(F¢) < dim(F) and that any element of F¢ is nilpotent.
5^°) Let d=dim(F). We assume that, for any finite dimensional vector space W on the field K, and any Lie algebra B ,made of nilpotent endomorphisms of W, and such that dim(B) £ d-1, there exists a non-vanishing vector x Î W such that u(x)=0 for any u Î B. We also keep the notations and hypothesis of question 4^°. Show that there exists a Lie algebra G₁,made of endomorphisms of V, obeying the following properties:

G Ì G1 Ì F

dim(G1) = dim(G) + 1

G1 is an ideal of G.

Conclude that there exists an ideal F₁ of F, such that dim(F₁)=d-1.
6^°) Proove Engel's theorem.
7^°) Show that any Lie algebra made of nilpotent operators of de the vector space V is triangulable.